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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#5-7">第5-7讲 一元函数微分学的应用</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1">1. 极值、单调性</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1-yx">（1）一元函数 <script type="math/tex">y(x)</script> 的极值</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2-fxy0-yx-yx">（2）一元函数隐函数 <script type="math/tex"> F(x,y)=0 </script> 确定 <script type="math/tex">y=(x)</script>，求 <script type="math/tex">y(x)</script> 的极值</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2">2. 拐点、凹凸性</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3">3. 渐近线</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_1">（1）水平渐近线和铅直渐近线</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2x">（2）斜渐近线的正确求法(在x趋向于无穷时)</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4">4. 曲率与曲率半径</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5">5. 中值定理</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_2">定理1（费马定理）</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_1">定理2（罗尔定理）</a>
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  <h1 id="数学-高等数学 3 第5-7讲 一元函数微分学的应用" class="content-subhead">数学-高等数学 3 第5-7讲 一元函数微分学的应用</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
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    <h2 id="5-7">第5-7讲 一元函数微分学的应用</h2>
<h3 id="1">1. 极值、单调性</h3>
<h4 id="1-yx">（1）一元函数 <script type="math/tex">y(x)</script> 的极值</h4>
<h4 id="2-fxy0-yx-yx">（2）一元函数隐函数 <script type="math/tex"> F(x,y)=0 </script> 确定 <script type="math/tex">y=(x)</script>，求 <script type="math/tex">y(x)</script> 的极值</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
F_x'+F_y'y_x'&=0 \\[1ex]
\Rightarrow
y_x'&=-\cfrac{F_x'}{F_y'}=0 \\[1ex]
\Rightarrow
F_x'&=0\ \ \ \ 得到极值点 \\[2em]
y_x''&=-\cfrac{F_{xx}''F_y'-F_x'F_{yy}''y_x'}{(F_y')^2} \\[1ex]
&=-\cfrac{F_{xx}''}{F_y'}\ \ \ \ 判断极大值极小值
\end{split}\end{equation}
</script>
</p>
<h3 id="2">2. 拐点、凹凸性</h3>
<ul>
<li>拐点：凹凸性改变的分界点</li>
<li>拐点存在的 <strong>必要条件</strong>：<script type="math/tex">f''(x_0)=0</script>
</li>
</ul>
<h3 id="3">3. 渐近线</h3>
<h4 id="1_1">（1）水平渐近线和铅直渐近线</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{x\to\infty}[f(x)] &= y_0,\ 则y=y_0为水平渐近线 \\[2ex]
实际上是求 \lim_{x\to\infty}[f(x)-y_0] &= 0 \\[1em]
\lim_{x\to x_0}[f(x)]&=\infty,\ 则x=x_0为铅直渐近线 
\end{split}\end{equation}
</script>
</p>
<h4 id="2x">（2）斜渐近线的正确求法(在x趋向于无穷时)</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{x\to\infty}(\cfrac{f(x)}{x}) &= A \\[1ex]
\lim_{x\to\infty}[f(x)-Ax] &= B \\[1ex]
渐近线方程为\ y &= Ax + B \\[1em]
实际上是求\ \lim_{x\to\infty}[f(x)-(Ax+B)] &= 0
\end{split}\end{equation}
</script>
</p>
<h3 id="4">4. 曲率与曲率半径</h3>
<p><img class="pure-img" alt="qulv" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/qulv.png" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
平均曲率：\overline k &= \bigg|\cfrac{\Delta \alpha}{\Delta s}\bigg| \\[1ex]
曲率：k &= \lim_{\Delta s \to 0}\bigg|\cfrac{\Delta \alpha}{\Delta s}\bigg| \\[1ex]
y' &= \tan\alpha \\[1ex]
y'' = \sec^2\alpha \cfrac{d\alpha}{dx} ⟺ d\alpha &= \cfrac{y''}{1+\tan^2\alpha}dx \\[1ex]
&= \cfrac{y''}{1+(y')^2}dx\\[1ex]
ds &= \sqrt{1+(y')}dx \\[1ex]
曲率：k&=\cfrac{|y''|}{[1+(y')^2]^{\frac{3}{2}}} \\[1em]
曲率半径：R&=\cfrac{1}{k}
\end{split}\end{equation}
</script>
</p>
<h3 id="5">5. 中值定理</h3>
<h4 id="1_2">定理1（费马定理）</h4>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学 第5-7讲 一元函数微分学的应用.assets/截屏2021-12-15 00.15.08.png" alt="截屏2021-12-15 00.15.08" style="zoom:33%;" /><br />
<script type="math/tex; mode=display">
设f(x)在x_0处满足
\begin{cases}
可导 \\
取极值
\end{cases}
，则f'(x)=0
</script>
</p>
<h4 id="2_1">定理2（罗尔定理）</h4>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学 第5-7讲 一元函数微分学的应用.assets/截屏2021-12-15 00.14.37.png" alt="截屏2021-12-15 00.14.37" style="zoom:33%;" /><br />
<script type="math/tex; mode=display">
设f(x)在x_0处满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导 \\
f(a) = f(b)
\end{cases}
，则存在\xi\in(a,b)，使得f'(\xi)=0
</script>
</p>
<h4 id="3_1">定理3（拉格朗日中值定理）</h4>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学 第5-7讲 一元函数微分学的应用.assets/截屏2021-12-15 00.13.45.png" alt="截屏2021-12-15 00.13.45" style="zoom:33%;" /><br />
<script type="math/tex; mode=display">
设f(x)在满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导
\end{cases}
，则存在\xi\in(a,b)，使得 \\
f(b) - f(a) = f'(\xi)(b - a) \\ 
即f'(\xi) = \cfrac{f(b) - f(a)}{b - a}
</script>
</p>
<h4 id="4_1">定理4（柯西中值定理）</h4>
<p>
<script type="math/tex; mode=display">
设f(x),g(x)在满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导 \\
g'(x)\ne0
\end{cases}
，则存在\xi\in(a,b)，使得 \\ 
\cfrac{f'(\xi)}{g'(\xi)} = \cfrac{f(b) - f(a)}{g(b) - g(a)}
</script>
</p>
<h4 id="5_1">定理5（泰勒中值定理）（泰勒公式 / 麦克劳林公式）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n \\[2em]
(带拉格朗日余项)\ \ f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - x_0)^{n+1} \\[2ex]
(皮亚诺余项)\ \ f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + o((x - x_0)^{n})
\end{split}\end{equation}
</script>
</p>
<h4 id="6">定理6（积分中值定理）</h4>
<p>
<script type="math/tex; mode=display">
设f(x)在[a,b]上连续，则存在\xi\in[a,b]，使得 \\ 
\int_{a}^{b}f(x)dx = f(\xi)(b - a)
</script>
</p>
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